Download Short-Period RF Undulator for a Nanometer SASE Source

Proceedings of IPAC2011, San Sebastián, Spain
THPC169
A SHORT-PERIOD RF UNDULATOR FOR A NANOMETER SASE SOURCE
S.V. Kuzikov#, A.A. Vikharev, M.E. Plotkin, Institute of Applied Physics, Nizhny Novgorod, Russia
J.L. Hirshfield, Omega-P, Inc., Yale University, New Haven, CT, USA
T.C. Marshall, Columbia University, New York, NY, USA
G.V. Sotnikov, NSC/KIPT, Kharkov, Ukraine
A.V. Peskov, Nizhny Novgorod State University, Nizhny Novgorod, Russia
An undulator made of static magnets typically has a period of a few centimeters. In order to produce synchrotron
radiation of short wavelengths (~ 1 nm) in such undulator
a 1.5 GeV electron beam is necessary. Instead of expensive high energetic electron beam a beam of a moderate
energy ( 300 MeV) in RF undulator with period ranging
from several millimeters to sub-millimeters is appealing.
Additionally the RF undulator can provide fast dynamic
control of polarization, wavelength, undulator K parameter, can have wide aperture (~ 1 cm), and to bring other
attractive features [1-3].
In order to become a competitive alternative to the traditional undulator, the RF undulator must, of course, have
as high strength of wiggling fields as magnetic fields in
existing permanent magnets (B ~ 1 T). In particular, the K
parameter should be about unity in RF undulator of the
10 mm period. Because powers of modern millimeter and
sub-millimeter RF sources like a magnicon or a relativistic gyroklystron are limited by several tens of megawatts,
necessary undulator fields can be obtained in so-called
storage cavities only. Design of such cavities should take
into account many constraints like: emittance growth in
presence of powerful RF fields, Ohmic wall losses,
breakdown, and surface pulse heating.
TRAVELLING WAVE RF UNDULATORS
A travelling wave undulator can be made of low-loss
resonant ring where an operating wave moves toward the
beam which traverses one of arms. Among low-loss
waves there are TE01 mode in smooth circular crosssection waveguide and HE11 in corrugated waveguide of
the impedance type. Both modes have low surface fields
in an oversized waveguide. Because the first mode requires bigger diameter to turn [4] and consequently requires higher RF power and energy, the HE11 mode is
preferable.
The 34 GHz room-temperature RF undulator which
exploits superposition of the HE11 and HE12 modes to
minimize diffraction losses in mitre bends is shown in
Fig. 1. Calculations show, in order to obtain parameter K
= 0.4 in the waveguide of L = 250 mm length, about
120 MW input power, 470 ns filling time (loaded Qfactor is 5.1·104 at waveguide radius 26.5 mm) are necessary. Surface electric field is ~ 20 MV/m. Ohmic quality
(Qohm = 31·104) here is twice more than diffraction quality (Qdiff = 15.2·104) caused by scattering in mitre bends.
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The mentioned high input power level is explained by big
total volume which cannot be reduced because waveguide
radius reduction causes more diffraction losses.
Figure 1: HE11 resonant ring.
BEAM DYNAMICS IN A STANDING
WAVE RF UNDULATOR
In the standing wave RF undulator an operating mode
consists of a co-propagating wave and a counter propagating wave (Fig. 2). Transverse oscillations in such undulator can be described by equation:
d2X K
 (    e i  e i )  e i ,
d 2

(1)
where X – is a dimensionless transverse particle position,
 – is a normalized longitudinal coordinate, K and  - are
undulator parameter and Lorentz factor,  - is an injection phase,  - is an amplitude of the co-propagating
wave. Here we used a definition:

k h
,
k h
(2)
where h – is a propagation constant in a waveguide, and
k=/c.
The counter propagating wave causes the desired FEL
quiver electron motion with relativistic Doppler up-shift
of Compton photons. The forward co-propagating wave
also interacts with the electrons and generates low frequency microwave radiation. Unfortunately, this wave
produces beam oscillations which by factor 1/ bigger
than oscillations caused by the counter propagating wave.
02 Synchrotron Light Sources and FELs
T15 Undulators and Wigglers
3293
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Abstract
THPC169
Proceedings of IPAC2011, San Sebastián, Spain
The view along the beam axis in Fig. 2, obtained by
means of Microwave CST Studio, shows such beam motion. The reason of this instability is that co-propagating
wave in oversized waveguide moves synchronously with
moving electrons far enough so that some electrons acquire the positioning shift in one direction, other electrons do the shift in the opposite direction.
Figure 2: Electron beam motion in standing wave undulator with identical co-propagating and counter propagating
waves (f = 34 GHz, K = 0.1,  = 13, R = 14 mm).
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The mentioned high-magnitude oscillations can cause
beam transportation problems and can cause undesirable
light power loss due to a fact that different parts of electron beam radiate power in different phases (incoherence). Criteria for maximum acceptable transverse beam
deflection are given by obvious Fresnel parameter:
smax
L

,
2
(3)
where  - is a wavelength of the deflecting RF radiation,
and L – is an undulator’s length. Note that increasing of
Lorentz factor  helps one to solve beam transportation
problem due to a reduction of a magnitude of oscillations.
However, equation (3) shows that acceptable average
beam deflection reduces linearly with  decreasing.
A CAVITY WITH DIFFERENT CO- AND
COUNTER PROPAGATING MODES
In order to avoid the mentioned undesirable instability
a co-propagating mode should have a transverse structure
which excludes an interaction with the beam. In Fig. 3 the
counter propagating mode is the TE01 wave which has
almost maximum fields at beam location, the copropagation mode TE02 has zero fields at this place (Fig.
4). Both modes do not have normal to wall fields.
(diffraction losses at level – 50 dB) is achievable. Input
coupler in a form of the perforated section couples selectively the TE02 mode in undulator with the TE01 coaxial
mode. Parameters of the 34 GHz undulator with K = 0.4
are summarized in the Table 1.
Figure 4: Cross-section of the resonator showing azimuthal transverse electric fields of co- and counter
propagating modes and a location of electron beam.
Table 1: Parameters of the 34 GHz Undulator
waveguide radius, R
12 mm
main section length, L
250 mm
Ohmic quality, Qohm
67800
loaded quality, QL
33900
coupling, 
0.8%
RF pulse duration, 
320 ns
beam shift off axis, rb
6.55 mm
RF input power, Pin
34 MW
temperature rise, T
133C
Beam dynamics in such undulator is described by solution of the equation (1) under =0:
X 
K

(e i  i  1)  e i ,
dX
K
 i (e i  1)  e i .
d

(4)
Analysis of equations (4) shows that there are injection
phases (=0 and =) which at length of undulator
end=2n, corresponded to an integer number of oscillations n, provide zero transverse position and velocity at
output simultaneously.
A CAVITY WITH A TM MODE NEAR CUT
OFF
Figure 3: Conceptual sketch of the two-mode undulator.
The TE01 and TE02 modes are converted each to other
by ladder-like converters in both ends of the undulator.
Calculations show that very high conversion efficiency
In this undulator the operating TM11 mode is excited in
barrel-like cavity (Fig. 5). Near cut off frequency both coand counter propagating modes have high phase velocities and can not be in synchronism with electron beam
long time. Beam dynamics simulations show that mentioned beam instability is avoided in this case. In cavity,
those length equals tens of wavelengths, an electric field
consists mainly of Ez component, so the surface field is
02 Synchrotron Light Sources and FELs
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T15 Undulators and Wigglers
Proceedings of IPAC2011, San Sebastián, Spain
not high. A corrugation is necessary, in order to provide
coupling of the TM11 mode with TE11 waves of two powering phase-locked RF sources. Polarization of the operating mode can be easily controlled by polarizations of
the input TE11 waves. The cavity has the smallest volume
in comparison with TM11 undulator where the operating
mode is far from cut off. This factor leads to relatively
small input power and input pulse duration. For example,
34 GHz undulator with K = 0.4 and effective length
250 mm (cavity radius R = 5.33 mm) requires Pin =
20 MW and  = 70 ns. These parameters provide low
surface electric field (~ 1 MV/m) and comfortable temperature rise (16C). A disadvantage of this scheme is
that frequency of output photons is proportional to 22 (
– is a Lorentz factor) like in a DC undulator instead of 42
like in RF undulators with far from cut off waves.
THPC169
with near to cut off TM11 mode has frequency up-shift
two times less in comparison with other considered versions , but it is very attractive due to many reasons (low
input power, low surface fields, simple design, and a possibility to be operated in a regime of long bunches.
Figure 6: Transverse position of electron at output of
50 cm long undulator section (end=100) as a function of
length and injection phase (f = 34 GHz, K = 0.4,  = 13).
It is remarkable that field structure in a cavity with near
to cut off TM11 mode has smooth growing part in the beginning and dropping part in the end. Such natural field
shaping allows provide small transverse shift and velocity
at output for arbitrary values of the injection phase as it is
shown in Figs. 6 and 7. This allows, in principle, to use
long electron bunches (longer than a half of wavelength)
or even to apply continuous beams.
“COLD” RF UNDULATOR
In order to reduce input power, it is natural to reduce
Ohmic losses by using cryogenically cooled systems.
Because a superconducting RF undulator is rather expensive and is able at relatively low RF frequencies  1 GHz,
it is attractive to use at Ka-band a copper undulator to be
cooled by liquid nitrogen (T80 C). In particular, the
copper cooling down to liquid nitrogen temperatures
promises more than 2 times loss reduction with anomalous skin effect taken into account. This predicted loss
reduction was demonstrated experimentally with 34 GHz
TE02 test cavity [5].
CONCLUSION
Among the considered schemes the HE11 resonant ring
is distinguished by simplest design, low surface fields,
but the feeding requires huge RF power. The TE01/TE02
resonator (off-axis beam) with short efficient mode converters and coaxial coupler requires a moderate power,
has low surface fields, in case of smoothly profiled RF
field it can be operated in a regime of long pulses, but
pulse heating temperature is rather high. The undulator
Figure 7: Transverse velocity of electron at output of
50 cm long undulator section (end=100) as a function of
length and injection phase (f = 34 GHz, K = 0.4,  = 13).
REFERENCES
[1] T. Shintake, K. Huke, J. Tanaka, I. Sato and
I. Kumabe, Development of microwave undulator,
Japanese Journal of Applied Physics 22 (1983) 844.
[2] G.G. Denisov et al., Int. Jouirnal of Infrared and Millimeter Waves, Vol. 5 Issue 10 (1984) 1389.
[3] S. Tantawi et al, Phys. Rev. ST Accel. Beams 8
(2005) 042002.
[4] A. Bogdashov,
G. Denisov
,D. Lukovnikov,
Y. Rodin, J. Hirshfield, Ka-band resonant ring for
testing components for a high-gradient linear accelerator, IEEE Trans. on Microwave Theory and Techniques, Vol. 53 Issue 10 (2005) 3152.
[5] J.L.Hirshfield, S.V. Kuzikov et al., A Short-Period
RF Undulator, Proc. of 8th Int. Workshop Strong microwaves and terahertz waves: sources and applications, Nizhny Novgorod, Russia, p. 93 (2011).
02 Synchrotron Light Sources and FELs
T15 Undulators and Wigglers
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c 2011 by IPAC’11/EPS-AG — cc Creative Commons Attribution 3.0 (CC BY 3.0)
Copyright ○
Figure 5: Conceptual design of the cavity with near to cut
off TM11 mode.